A function f(x) is defined by: f(x) = { m * sinx + n, if x <= 2 pi; x^2 - pix + 5, if x > 2 pi.

Question

a. Given that f(x) is a differentiable function, find the values of m and n.

b.Write the equation of the tangent line to f(x) at x = 2pi.

c. Use the tangent line equation to find an approximation for the value of f(6). You may leave your answer in terms of pi.

Please show work, all help is greatly appreciated.

Answer 1

so, at x = 2π, we need and y' to match:
m sinx + n = x^2 - πx + 5
m cosx = 2x - π + 5
That is,
m*0 + n = 4π^2 - 2π^2 + 5
n = 2π^2 + 5
and
m = 4π-π = 3π
So, for x <= 2π, y = 3π sinx + 2π^2 + 5

See what you can do now with (b) and (c)